computer vision - a modern approach set: recognition as template matching slides by d.a. forsyth...

Computer Vision - A Modern ApproachSet: Recognition as Template Matching

Slides by D.A. Forsyth

Recognition by finding patterns

• We have seen very simple template matching (under filters)

• Some objects behave like quite simple templates – Frontal faces

• Strategy:– Find image windows

– Correct lighting

– Pass them to a statistical test (a classifier) that accepts faces and rejects non-faces

Basic ideas in classifiers

• Loss– some errors may be more expensive than others

• e.g. a fatal disease that is easily cured by a cheap medicine with no side-effects -> false positives in diagnosis are better than false negatives

– We discuss two class classification: L(1->2) is the loss caused by calling 1 a 2

• Total risk of using classifier s

Basic ideas in classifiers

• Generally, we should classify as 1 if the expected loss of classifying as 1 is better than for 2

• gives

• Crucial notion: Decision boundary– points where the loss is the same for either case

Some loss may be inevitable: the minimumrisk (shaded area) is called the Bayes risk

Finding a decision boundary is not the same asmodelling a conditional density.

• Assume normal class densities, p-dimensional measurements with common (known) covariance and different (known) means

• Class priors are• Can ignore a common factor in

posteriors - important; posteriors are then:

p x k( ) =12π ⎛ ⎝

⎞ ⎠

Σ−1

2exp−

12x−μ

k( )TΣ−1 x−μ

k( ) ⎡ ⎣

⎤ ⎦

Example: known distributions

p k | x( )∝ π k( )1

2π ⎛ ⎝

⎞ ⎠

− p2

Σ−1

2 exp −12x − μ

Σ−1 x − μk( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

• Classifier boils down to: choose class that

minimizes:

δ x, μk( )

2− 2 logπ k

δ x, μk( ) = x − μ

Σ −1 x− μk( )

⎡ ⎣

⎤ ⎦

because covariance is common, this simplifies to sign ofa linear expression:

Mahalanobis distance

Plug-in classifiers

• Assume that distributions have some parametric form - now estimate the parameters from the data.

• Common: – assume a normal distribution with shared covariance, different means; us

usual estimates– ditto, but different covariances; ditto

• Issue: parameter estimates that are “good” may not give optimal classifiers.

picture from Ripley

Histogram based classifiers

• Use a histogram to represent the class-conditional densities– (i.e. p(x|1), p(x|2), etc)

• Advantage: estimates become quite good with enough data!

• Disadvantage: Histogram becomes big with high dimension– but maybe we can assume feature independence?

Finding skin

• Skin has a very small range of (intensity independent) colours, and little texture– Compute an intensity-independent colour measure, check if colour

is in this range, check if there is little texture (median filter)– See this as a classifier - we can set up the tests by hand, or learn

them.– get class conditional densities (histograms), priors from data

(counting)

• Classifier is

Finding faces

• Faces “look like” templates (at least when they’re frontal).

• General strategy:– search image windows at a

range of scales

– Correct for illumination

– Present corrected window to classifier

• Issues– How corrected?

– What features?

– What classifier?

– what about lateral views?

Naive Bayes

• (Important: naive not necessarily perjorative)• Find faces by vector quantizing image patches, then

computing a histogram of patch types within a face• Histogram doesn’t work when there are too many features

– features are the patch types– assume they’re independent and cross fingers– reduction in degrees of freedom– very effective for face finders

• why? probably because the examples that would present real problems aren’t frequent.

Many face finders on the face detection home pagehttp://home.t-online.de/home/Robert.Frischholz/face.htm

Face Recognition

• Whose face is this? (perhaps in a mugshot)

• Issue:– What differences are important

and what not?

– Reduce the dimension of the images, while maintaining the “important” differences.

• One strategy:– Principal components analysis

• Many face recognition strategies at http://www.cs.rug.nl/users/peterkr/FACE/face.html

Difficulties with PCA

• Projection may suppress important detail– smallest variance directions may not be unimportant

• Method does not take discriminative task into account– typically, we wish to compute features that allow good

discrimination

– not the same as largest variance

Linear Discriminant Analysis

• We wish to choose linear functions of the features that allow good discrimination.– Assume class-conditional covariances are the same

– Want linear feature that maximises the spread of class means for a fixed within-class variance

Neural networks

• Linear decision boundaries are useful– but often not very powerful

– we seek an easy way to get more complex boundaries

• Compose linear decision boundaries– i.e. have several linear classifiers, and apply a classifier to their

output

– a nuisance, because sign(ax+by+cz) etc. isn’t differentiable.

– use a smooth “squashing function” in place of sign.

Training

• Choose parameters to minimize error on training set

• Stochastic gradient descent, computing gradient using trick (backpropagation, aka the chain rule)

• Stop when error is low, and hasn’t changed much

Error p( ) =12 ⎛ ⎝

⎞ ⎠n xe; p( ) − oe( )

The vertical face-finding part of Rowley, Baluja and Kanade’s systemFigure from “Rotation invariant neural-network based face detection,” H.A. Rowley, S. Baluja and T. Kanade, Proc. Computer Vision and Pattern Recognition, 1998, copyright 1998, IEEE

Histogram equalisation gives an approximate fix for illumination induced variability

Architecture of the complete system: they use another neuralnet to estimate orientation of the face, then rectify it. They search over scales to find bigger/smaller faces.

Convolutional neural networks

• Template matching using NN classifiers seems to work

• Natural features are filter outputs– probably, spots and bars, as in texture

– but why not learn the filter kernels, too?

A convolutional neural network, LeNet; the layers filter, subsample, filter,subsample, and finally classify based on outputs of this process.

LeNet is used to classify handwritten digits. Notice that the test error rate is not the same as the training error rate, becausethe test set consists of items not in the training set. Not all classification schemes necessarily have small test error when theyhave small training error.

Support Vector Machines

• Neural nets try to build a model of the posterior, p(k|x)

• Instead, try to obtain the decision boundary directly– potentially easier, because we need to encode only the geometry of

the boundary, not any irrelevant wiggles in the posterior.

– Not all points affect the decision boundary

• Linearly separable data means

• Choice of hyperplane means

• Hence distance

Actually, we construct a dual optimization problem.

By being clever about what x means, I can have muchmore interesting boundaries.

Space in which decisionboundary is linear - aconic in the original spacehas the form

x, y( ) → x2 , xy,y2, x,y( ) = u0 ,u1,u2 ,u3 ,u4( )

au0 +bu1 + cu2 + du3 + eu4 + f = 0

Vision applications

• Reliable, simple classifier, – use it wherever you need a

classifier

• Commonly used for face finding

• Pedestrian finding– many pedestrians look like

lollipops (hands at sides, torso wider than legs) most of the time

– classify image regions, searching over scales

– But what are the features?– Compute wavelet coefficients

for pedestrian windows, average over pedestrians. If the average is different from zero, probably strongly associated with pedestrian

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